A normalizing-flow-based implementation of the density-of-states approach has recently been used to successfully reconstruct the partition function of (1+1)D scalar lattice field theory. In this preliminary work, we extend this framework to a lattice gauge theory by employing gauge-equivariant normalizing flows to reconstruct the density of states of pure (1+1)D U(1) lattice gauge theory, both with and without a $\theta$-term. In the absence of a $\theta$-term, we first demonstrate that the normalizing-flow-based reconstruction of the density of states reproduces the known analytic results for this theory. We further show that, in the presence of a $\theta$-term, this formulation enables the generation of gauge-field configurations at fixed values of the topological charge.